Fit Difference Between Nonnested Models Given Categorical Data: Measures and Estimation

In structural equation modeling (SEM) studies with categorical data, researchers often use the root mean square error of approximation (RMSEA), comparative fit index (CFI), or standardized root mean squared residual (SRMR) to compare rival models. Model selection based on ΔRMSEA, ΔCFI, or ΔSRMR is meaningful because (a) finding a better model is more scientific and easier than establishing a “good” model, (b) it avoids the problems with cutoffs for fit indices, (c) one is less likely to overlook other equally substantively plausible models, and (d) information criteria (e.g., AIC, BIC) are not applicable to categorical data SEM. In this paper, we propose point estimators and confidence intervals (CIs) for ΔRMSEA, ΔCFI, and ΔSRMR under categorical data. Our methods are applicable to nonnested models and do not need a true model. Simulation results show our point estimators and CIs are all trustworthy, whereas the bias is large when estimating ΔRMSEA (ΔCFI, ΔSRMR) based on the common estimators in the current literature for RMSEA (CFI, SRMR).